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Chapter 2 Automatic Lens Design; Managing the Lens Design Program 2.1 The Merit Function What is usually referred to as automatic lens design is, of course, nothing of the sort. The computer programs which are so described are actually optimization programs which drive an optical design to a local optimum, as defined by a merit function which is not a true merit function, but actually a defect function. In spite of the preceding disclaimers, we will use these commonly accepted terms in the discussions which follow. Broadly speaMng, the merit function can be described as a combination or function of calculated characteristics, which is intended to completely describe, with a single number, the value or quality of a given lens design. This is obviously an exceedingly difficult thing to do. The typical merit function is the sum of the squares of many image defects; usually these image defects are uated for three locations in the field of view unless the system covers a very large or a very small angular field. The squares of the defects are used so that a negative value of one defect does not offset a positive value of some other defect. The defects may be of many different kinds; usually most are related to the quality of the image. However, any characteristic which can be calculated may be assigned a target value and its departure from that target regarded as a defect. Some less elaborate programs utilize the third-order Seidel aberrations; these provide a rapid and efficient way of adjusting a design. These cannot be regarded as optimizing the image quality, but they do work well in correcting ordinary lenses. Another type of merit function traces a large number of 3 4 Chapter Two rays from an object point. The radial distance of the image plane intersection of the ray from the centroid of all the ray intersections is then the image defect. Thus the merit function is effectively the sum of the root-mean-square rms spot sizes for several field angles. This type of merit function, while inefficient in that it requires many rays to be traced, has the advantage that it is both versatile and in some ways relatively foolproof. Some merit functions calculate the values of the classical aberrations, and convert or weight them into their equivalent wavefront deations. See ulary Sec. F-12 for the conversion factors for several common aberrations. This approach is very efficient as regards computing time, but requires careful design of the merit function. Still another type of merit function uses the variance of the wavefront to define the defect items. The merit function used in the various David Grey programs is of this t3rpe, and is certainly one of the best of the commercially available merit functions in producing a good balance of the aberrations. Characteristics which do not relate to image quality can also be controlled by the lens design program. Specific construction parameters, such as radii, thicknesses, spaces, and the like, as well as focal length, working distance, magnification, numerical aperture, required clear apertures, etc., can be controlled. Some programs include such items in the merit function along with the image defects. There are two drawbacks which somewhat offset the neat simplicity of this approach. One is that if the first-order characteristics which are targeted are not initially close to the target values, the program may correct the image aberrations without controlling these first-order characteristics; the result may be, for example, a well-corrected lens with the wrong focal length or numerical aperture. The program often finds this to be a local optimum and is unable to move away from it. The other drawback is that the inclusion of these items in the merit function has the effect of slowing the process of improving the image quality. An alternative approach is to use a system of constraints outside the merit function. Note also that many of these items can be controlled by features which are included in almost all programs, namely angle-solves and height-solves. These algebraically solve for a radius or space to produce a desired ray slope or height. In any case, the merit function is a summation of suitably weighted defect items which, it is hoped, describes in a single number the worth of the system. The smaller the value of the merit function, the better the lens. The numerical value of the merit function depends on the construction of the optical system; it is a function of the construction parameters which are designated as variables. Without getting into the details of the mathematics involved, we can realize that the merit function is an n-dimensional space, where n is the number of the vari Automatic Lens Design 5 able constructional parameters in the optical system. The task of the design program is to find a location in this space i.e., a lens prescription or a solution vector which minimizes the size of the merit function. In general, for a lens of reasonable complexity there will be many such locations in a typical merit function space. The automatic design program will simply drive the lens design to the nearest minimum in the merit function. 2.2 Optimization The lens design program typically operates this way Each variable parameter is changed one at a time by a small increment whose size is chosen as a compromise between a large value to get good numerical accuracy and a small value to get the local differential. The change produced in every item in the merit function is calculated. The result is a matrix of the partial derivatives of the defect items with respect to the parameters. Since there are usually many more defect items than variable parameters, the solution is a classical least- squares solution. It is based on the assumption that the relationships between the defect items and the variable parameters are linear. Since this is usually a false assumption, an ordinary least-squares solution will often produce an unrealizable lens or one which may in fact be worse than the starting design. The damped least-squares solution, in effect, adds the weighted squares of the parameter changes to the merit function, heavily penalizing any large changes and thus limiting the size of the changes in the solution. The mathematics of this process are described in Spencer, “A Flexible Automatic Lens Correc- tion Program,n Applied Optics, vol. 2, 1963, pp. 1257-1264, and by Smith in W. Driscoll ed., Handbook of Optics, McGraw-Hill, New York, 1978. If the changes are small, the nonlinearity will not ruin the process, and the solution, although an approximate one, will be an improvement on the starting design. Continued repetition of the process will eventually drive the design to the nearest local optimum. One can visualize the situation by assuming that there are only two variable parameters. Then the merit function space can be compared to a landscape where latitude and longitude correspond to the variables and the elevation represents the value of the merit function. Thus the starting lens design is represented by a particular location in the landscape and the optimization routine will move the lens design downhill until a minimum elevation is found. Since there may be many depressions in the terrain of the landscape, this optimum may not be the best there is; it is a local optimum and there can be no as- surance except in very simple systems that we have found a global 6 Chapter Two optimum in the merit function. This simple topological analogy helps to understand the dominant limitations of the optimization process the program finds the nearest minimum in the merit function, and that minimum is uniquely determined by the design coordinates at which the process is begun. The landscape analogy is easy for the human mind to comprehend; when it is extended to a 10- or 20- dimension space, one can realize only that it is apt to be an extremely complex neighborhood. 2.3 Local Minima Figure 2.1 shows a contour map of a hypothetical two-variable merit function, with three significant local minima at points Ay By and C; there are also three other minima at D, E, and F. It is immediately apparent that if we begin an optimization at point Z, the minimum at point B is the only one which the routine can find. A start at Y on the ridge at the lower left will go to the minimum at C. However, a start Figure 2.1 Topography of a hypothetical two-variable merit function, with three significant minima A, B, C and three trivial minima D, Ey F. The minimum to which a design program will go depends on the point at which the optimization process is started. Starting points X, Y} and Z each lead to a different design minimum; other starting points can lead to one of the trivial minima. Automatic Lens Design 7 at X} which is only a short distance away from Y} will find the best minimum of the three, at point A. If we had even a vague knowledge of the topography of the merit function, we could easily choose a starting point in the lower right quadrant of the map which would guarantee finding point A. Note also that a modest change in any of the three starting points could cause the program to stagnate in one of the trivial minima at D3 Et or F. It is this sort of minimum from which one can escape by “jolting” the design, as described below. The fact that the automatic design program is severely limited and can find only the nearest optimum emphasizes the need for a knowledge of lens design, in order that one can select a starting design which is close to a good optimum. This is the only way that an automatic program can systematically find a good design. If the program is started out near a poor local optimum, the result is a poor design. The mathematics of the damped least-squares solution involves the inversion of a matrix. In spite of the damping action, the process can be slowed or aborted by either of the following conditions 1 A variable which does not change or which produces only a very small change in the merit function items. 2 Two variables which have the same, nearly the same, or scaled effects on the items of the merit function. Fortunately, these conditions are rarely met exactly, and they can be easily avoided. If the program settles into an unsatisfactory optimum such as those at D, E} and F in Fig. 2.1 it can often be jolted out of it by manually introducing a significant change in one or more parameters. The trick is to make a change which is in the direction of a better design . Again, a knowledge of lens designs is virtually a necessity. Sometimes simply freezing a variable to a desirable can be sufficient to force a move into a better neighborhood. The difficulty is that too big a change may cause rays to miss surfaces or to encounter total internal reflection, and the optimization process may break down. Con- versely, too small a change may not be sufficient to allow the design to escape from a poor local optimum. Also, one should remember that if the program is one which adjusts optimizes the damping factor, the factor is usually made quite small near an optimum, because the program is taking small steps and the situation looks quite linear; after the system is jolted, it is probably in a highly nonlinear region and a big damping factor may be needed to prevent a breakdown. A manual increase of the damping factor can often avoid this problem. Another often-encountered problem is a design which persists in moving to an obviously undesirable when you know that there is a mudi better, very different onethe one that you want. Freezing the of one part of the lens for a few cycles of optimization will often allow the rest of the lens to settle into the neighborhood of the 8 Chapter Two desired optimum. For example, if one were to try to convert a Cooke triplet into a split front crown , the process might produce either a which is like the original triplet with a narrow airspaced crack in the front crown, or a with rather wild meniscus elements. A technique which will usually avoid these unfortunate local optima in this case is to freeze the front element to a plano-convex by fixing the second surface to a plane for a few qycles of optimization. Again, one must know which lens s are the good ones. 2.3 Types of Merit Functions Many programs allow the user to define the merit function. This can be a valuable feature because it is almost impossible to design a truly universal merit function. As an example, consider the design of a simple Fraunhofer telescope objective a merit function which controls the spherical and chromatic aberrations of the axial marginal ray and the coma of the oblique ray bundle plus the focal length is all that is necessary. If the design complexity is increased by allowing the airspace to vary and/or adding another element, the merit function may then profitably include entries which will control zonal spherical, spherochromatism, and/or fifth-order coma. But as long as the lens is thin and in contact with the aperture stop, it would be foolish to include in the merit function entries to control field curvature and astigmatism. There is simply no way that a thin stop-in-contact lens can have any control over the inherent large negative astigmatism; the presence of a target for this aberration in the merit function will simply slow down the solution process. It would be ridiculous to use a merit function of the type required for a photographic objective to design an ordinary telescope objective. Indeed, an attempt to correct the field curvature may lead to a eomprdmise design with a severely undercorrected axial spherical aberration which, in combination with coma, may fool the computer program into thiiiking that it has found a useful optimum. There are many design tasks in this category, where the require- ments are effectively limited in number and a simple, equally limited merit function is clearly the best choice. In such cases, it is usually obvious that some specific state of correction will yield the best results; there is no need to balance the correction of one aberration against another. More often, however, the situation is not so simple; compromises and balances are required and a more complex, suitably weighted merit function is necessary. This can be a delicate and somewhat tricky matter. For example, in the design of a lens with a significant aperture and field, there is almost always a poor local optimum in Automatic Lens Design 9 which 1 the spherical aberration is left quite undereorrected，（ 2 a compromise focus is chosen well inside the paraxial focus, 3 the Petzval field is made inward-curving, and 4 overcorrected oblique spherical aberration is introduced to “balance” the design. A program which relies on the rms spot radius for its merit function is very likely to fall into this trap. A better d